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Quantized form factor shift in the presence offree electron laser radiation

F. Fratini, L. Safari, A. G. Hayrapetyan, K. Jankala,

P. Amaro and J. P. Santos

EPL, 107 (2014) 13002

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EPL, 107 (2014) 13002 www.epljournal.org

doi: 10.1209/0295-5075/107/13002

Quantized form factor shift in the presence of free electron laser

radiation

F. Fratini1,2,3(a), L. Safari2, A. G. Hayrapetyan4,5, K. Jankala2, P. Amaro4,6 and J. P. Santos6

1 Universidade Federal de Minas Gerais, Instituto de Ciencias Exatas, Departamento de Fısica

31270-901 Belo Horizonte, MG, Brasil2 Department of Physics, University of Oulu - Fin-90014 Oulu, Finland3 Institut Neel-CNRS - BP 166, 25 rue des Martyrs, F-38042 Grenoble Cedex 9, France4 Physikalisches Institut, Ruprecht-Karls-Universitat Heidelberg - D-69120 Heidelberg, Germany5 Max-Planck-Institut fur Physik komplexer Systeme - D-01187 Dresden, Germany6 Centro de Fısica Atomica, Departamento de Fısica, Faculdade de Ciencias e Tecnologia, FCT,

Universidade Nova de Lisboa - P-2829-516 Caparica, Portugal

received 16 March 2014; accepted in final form 13 June 2014published online 7 July 2014

PACS 34.80.Qb – Laser-modified scatteringPACS 34.80.Bm – Elastic scatteringPACS 87.64.Bx – Electron, neutron and x-ray diffraction and scattering

Abstract – In electron scattering, the target form factors contribute significantly to the diffractionpattern and carry information on the target electromagnetic charge distribution. Here we showthat the presence of electromagnetic radiation, as intense as currently available in free electronlasers, shifts the dependence of the target form factors by a quantity that depends on the number ofphotons absorbed or emitted by the electron as well as on the parameters of the electromagneticradiation. As example, we show the impact of intense ultraviolet and soft X-ray radiation onelastic electron scattering by the Ne-like argon ion and by the xenon atom. We find that the shiftbrought by the radiation to the form factor is of the order of some percent. Our results may openup a new avenue to explore matter with the assistance of laser.

Copyright c© EPLA, 2014

Introduction. – Electron scattering is a tool of greatimportance for exploring the structure of matter [1–6].The most significant example to highlight such impor-tance is perhaps electron microscopy, where electron scat-tering is the core process [7]. Electron scattering can beelastic or inelastic. In both cases, the diffraction pat-tern of the scattered electrons is influenced by the tar-get form factor (FF), also called scattering factor [8,9].Both elastic and inelastic FFs carry information aboutthe electromagnetic charge distribution of the target [8,9].FFs are not only relevant in electron scattering, but playan important role also in light diffraction [10] and lightabsorption [11]. As a consequence, FFs are the sub-ject of research in many scientific areas, such as atomicphysics [12], nuclear and subnuclear physics [13,14], crys-tallography [15] and biology [16,17]. Due to the wide ap-plicability of FFs, exploring the effects of electromagnetic

(a)E-mail: fratini.filippo@gmail.com;ffratini@fisica.ufmg.br

radiation (ER) on the target FFs is of general interest inscience.

Laser-assisted electron scattering (LAES) has been thesubject of intense research since the 1970s, as a conse-quence of the development of lasers [18–21]. In this let-ter we show that the presence of intense ER in electronscattering can be used to control the FFs. More specif-ically, we show that the presence of ER shifts the argu-ment of the inelastic and elastic FFs from Q to Q + sk,where Q is the momentum transfer between electron andtarget, k is the wave vector of the ER, is the reducedPlanck constant, and s is an integer number that repre-sents the number of photons absorbed (if s > 0) or emitted(if s < 0) by the electron during the scattering process.A similar linear momentum shift on the whole differentialcross-section (DCS) has been already highlighted in theliterature [22–25]. However, to the best of our knowledge,the linear momentum shift brought by the ER to elasticand inelastic FFs has not been clearly pointed out in the

13002-p1

F. Fratini et al.

literature. In view of recent advances in X-ray free electronlaser (FEL) sources, it is important to highlight that ex-ploring FF shifts has become nowadays feasible with thecurrent state-of-the-art-technology. Quantized FF shiftscan be used to tailor the information extracted from thetarget electromagnetic distribution. More generally, quan-tized FF shifts can be used broadly in science, since FFsare used in various scientific areas. As examples, in thisletter we show the effect of intense ultraviolet and soft X-ray radiation on the DCS and the FF for elastic electronscattering by Ne-like argon and xenon atom. Our resultsfor the DCS are compared with measurements carried outwithout laser assistance. The shift brought to the FF bythe radiation is of the order of some percent.

Coherent X-ray light with high intensity, which hasbecome recently available due to FEL sources [26–28],can be used jointly with the theoretical formalism pre-sented in this letter to explore quantized shifts in FFsof mesoscopic, atomic and also nuclear targets. More-over, our results open up the possibility to measure FFvalues by varying the linear momentum of the ER (k),while keeping fixed the electron-target linear momentumtransfer (Q). This allows, for example, to measure FFsat zero argument in scattering events for which theelectron-target linear momentum transfer is not zero.The elastic FF at zero argument, although it carriesimportant information on the quadratic mean radiusof spherically symmetric charge distributions [9,29], isnormally difficult to measure in the absence of ER, sincein that case the signal of scattering events is suppressedby the background of non-scattering events.

Finally, experiments on free electron scattering by ionshave recently attracted much interest due to their straight-forward applications to plasma and astrophysics [30,31].Understanding the ER effects on the pattern of the scat-tered electrons and on the target FFs might thus havedirect impact on plasma diagnostics and astrophysics.

SI units are used throughout this letter.

Electron state inside the radiation. – Let us con-sider linearly polarized ER whose vector potential can betaken in the form A = A0 sin(k · r − ωt). Here, ω is theangular frequency while k is the wave vector of the ER(c|k| = ω, c is the speed of light in vacuum). Withinthe Coulomb gauge and by retaining only terms linear inω/(mc2), the free electron state inside the radiation canbe described by the wave function [32,33]

ΨP ,k(r, t) =e

i(P ·r−Et)

√V

exp[

iα(

1 − cos(k · r − ωt))]

× exp

[

−iβ

(

k · r − ωt − 1

2sin

(

2(k · r − ωt))

)]

, (1)

where V is the volume where the wave function is defined,α = eP · A0

/(

(P · k − mω))

, while β = e2A20

/(

4(P ·k−mω)

)

, and m and e are the electron mass and charge,respectively. The continuous quantum numbers P and E

in (1) are conserved quantities used to label the electronstate inside the ER. However, they do not represent thelinear momentum and the energy of the electron (which,on the other hand, are not conserved quantities), as longas the ER is on. They merely identify the electron stateinside the ER and are called “linear momentum” and “en-ergy” parameters [32]. Nevertheless, if the ER is switchedoff, the wave function (1) will become a plane wave and,consequently, P and E will then correctly represent theconserved linear momentum and energy of the electron, re-spectively. In either cases the ER is on or off, the energy-momentum relation 2mE = P 2 must be satisfied [34].Finally, we notice that the parameters α and β containthe dependence on the angle between the linear momen-tum parameter (P ) and the ER wave vector (k).

Scattering differential cross-section. – Let us nowconsider LAES by some target characterized by an ex-tended charge distribution and let us denote by θ the scat-tering angle. The electron-target interaction is assumed tobe electrostatic and the space where the scattering hap-pens is assumed to be filled with linearly polarized ER.We suppose that the ER be non-invasive, i.e., that thetarget is not perturbed by the ER, as usual in LAES [18].The amplitude for this scattering process can be writtenin first-order time-dependent perturbation theory as [34]

A = (i)−1e

∫ t

−t

dt′∫

d3r Ψ∗Pf ,k(r, t′)V (r) ΨPi,k(r, t′),

(2)where ∆t = 2t is the scattering time interval and the po-tential V (r) is of the form [8,9,35]

V (r) =

∫

d3rξ φ∗f (rξ)Vp

(

|r − rξ|)

φi(rξ). (3)

Here φi,f are the initial and final target wave functions,while Vp

(

|r − rξ|)

denotes the electrostatic potential be-tween two point-like charges at distance |r − rξ|. Wehave here assumed, for simplicity, that the target canbe described with a single variable rξ: The extension tomany-body targets is trivial. The amplitude (2) does notfully take into account the bound structure of the tar-get. Rather, the target is treated as a source of potential.The second-order scattering term should be included inorder to fully account for the target bound structure andresonances. Consequently, the validity range of (2) is re-stricted to electron energies far from target resonances.

The scattering amplitude (2) may be further manipu-lated by employing the Jacobi-Anger expansion of the ex-ponentials. The evaluation of the amplitude A proceedsstraightforwardly and is similar to the standard evalua-tion in the absence of ER, as showed, for example, inrefs. [8,9]. Mathematical details are given in the sup-plementary material [36]. On squaring the amplitude,extending the scattering time interval to infinity for ob-taining energy conservation, multiplying by the density offinal states and normalizing to the electron flux, we obtain

13002-p2

Quantized form factor shift in the presence of free electron laser radiation

the differential cross-section for electron scattering in thepresence of linearly polarized ER as

dσs

dΩ(Pi,Pf ) =

|Pf ||Pi|

J 2s (αi − αf )

(

2m c αe

(Q + sk)2

)2

×∣

∣F (Q + sk)∣

∣

2, (4)

where Js is the Bessel function of order s, Ef = Ei +sω = P 2

f /(2m) is the energy of the outgoing electron,Q = Pi − Pf is, as mentioned above, the linear momen-tum transfer between electron and target, s is an integernumber ranging from −∞ to +∞ representing the pho-ton number. In addition, αi(f) is the previously definedquantity α calculated for P → Pi(f), while αe is the elec-tromagnetic coupling constant.

The quantity F (Q) is the target FF and reads

F (Q) =

∫

d3r ei

r·Q φ∗f (r)φi(r). (5)

In elastic scattering, we have φi = φf , and, consequently,the FF represents the Fourier transform of the targetcharge distribution. In this case, the FF is referred to aselastic FF. On the other hand, if the scattering is inelas-tic, we have φi = φf , and the FF is referred to as inelasticFF [8]. In deriving eq. (4), we have furthermore assumedβi − βf ≈ 0, which is justified within the non-relativisticassumption. Here βi(f) is the previously defined quantityβ calculated for P → Pi(f).

In the case the target is an isolated atom, the scatteringcross-section gets contributions from both the nucleus andthe atomic electrons, for which the relative form factorsadd coherently but with opposite sign. For non-relativisticenergies, we may set the nuclear form factor equal to thenuclear charge (Zn). Therefore, in this case eq. (4) reads

dσs

dΩ(Pi,Pf ) =

|Pf ||Pi|

J 2s (αi − αf )

(

2m c αe

(Q + sk)2

)2

×∣

∣Zn − FAt(Q + sk)∣

∣

2, (6)

where FAt(Q) is the atomic form factor. If the ERis switched off (which is accomplished by settingαi,f ,k → 0), eq. (6) turns out to be equal to the Mottformula [8], as expected.

Since eq. (4) has been derived in non-relativistic first-order perturbation theory, it is not directly applicablewhen the electron energy is relativistic (Ei,f 100 keV) orwhen the scattering angle is particularly small (θ 2).However, relativistic, screening and spin corrections canbe added in a similar way as done for the scattering cross-section in the absence of ER, so as to make it applicable fora wider energy range and for small scattering angles [7,9].

By analyzing eq. (4), we notice that the energies of thescattered electrons turn out to be spanned by multiples ofthe photons energy. The integer number s is thus easilyinterpreted as the number of photons absorbed (if s > 0)or emitted (if s < 0) by the electron during the scatter-ing process, as normally done in LAES experiments [19].

Most importantly, we notice that the presence of ER shiftsthe argument of the FF from Q to Q + sk. The photonnumber s, on which the shift depends, can be measuredby detecting the kinetic energy of the scattered electron.Due to the fact that such shift does not depend on the ERpolarization, it will hold for any kind of ER polarizationand, also, for unpolarized light.

It must be underlined that eq. (4) is different fromeq. (210) of ref. [20], as our Q corresponds to QN inref. [20].

Kroll-Watson formula re-visited. – The Kroll-Watson formula (KWF) can be obtained either within theBorn approximation, which is in line with our derivation,or within the low-frequency limit (ω ≪ eP ·A0/m) [37].In the light of this, we expect to recover the KWF as aspecial case of our theory, in the case of elastic scattering,within the assumptions of the first Born approximation.Indeed, if the momentum carried by the absorbed or emit-ted photons is much lower than the momentum transferbetween electron and target, we may use Q + sk ≈ Q.Furthermore, the non-relativistic assumption allows forthe replacement Pi,f ·k−mω ≈ −mω in the denominatorsof αi,f . Employing these two approximations in (4) yieldsthe KWF for the differential cross-section

dσs

dΩ(Pi,Pf ) ≃ |Pf |

|Pi|J2

s (η)dσel

dΩ(Q), (7)

where η = −eQ · A0/(ωm). Kroll and Watson obtainedeq. (7) by analyzing the Green function that governs theLAES process. Our result (4) can be thus considered arefinement of the KWF which allows to grasp the effectthat the ER brings to the target FF.

The KWF has been widely investigated in atomicphysics and has been found to adequately describe thedata in several experimental cases (see ref. [19] for a briefaccount). However, it has been showed that the KWFis unable to describe LAES when the scattering angle issmall (θ ≃ 9) [38] and when the laser polarization is or-thogonal to the linear momentum transfer (A0 ⊥ Q) [39].Although a few attemps to solve the discrepancies havebeen made [40–43], the problem is still open [44]. On ac-count of this, we ought to say that the validity of (4) mayshare the same limitations of KWF.

Atomic examples. – Equation (4) is the main resultof this letter and describes the differential cross-sectionfor LAES. Specifically, we read from eq. (4) that, in thepresence of ER, the target FFs and the differential cross-section depend not only on the momentum transfer Q, butalso on the ER wavelength and direction, as well as on thenumber of photons absorbed or emitted during the scatter-ing process. Thus, these three last parameters can be suit-ably tuned, in experiments, so as to alter the shape of thetarget FF to be measured, for a given Q. In the following,we discuss two examples to better highlight the impact ofER on the DCS and on the target FFs.

13002-p3

F. Fratini et al.σ

ω

θ

Fig. 1: (Color online) Differential cross-section for elastic elec-tron scattering by Ne-like argon in the presence of intense ul-traviolet ER with wavelength ≃ 224 nm. The electron initialenergy is Ei = 22.46 eV. The ER has direction orthogonalto the scattering axis, linear polarization parallel to the lin-ear momentum transfer, and intensity I ≃ 1015 W/cm2. Thegreen short-dashed line refers to electron scattering in the ab-sence of ER, while the crosses represent the best values fromexperimental measurements taken in absence of ER [45]. Thecurves denoted by KW are obtained with KWF. s representsthe number of photons absorbed (if s > 0) or emitted (if s < 0)by the scattered electron.

Let us consider elastic electron scattering by an isolatedNe-like argon ion, in the presence of intense ultraviolet ERof wavelength ≃ 224 nm (the wavelength of HeAg lasers).In this energy range, the ER may not excite the target,since its first excitation line is at ∼ 4.95 nm. The targetis thus unaffected by the ER, as hypothesized. In fig. 1,we show the differential cross-section, dσs/dΩ, for sucha process, where electron energy and ER parameters arespecified. The experimental data agree well with the the-oretical predictions for angles θ 120. The theoreticalpredictions are given by eq. (6) with αi,f ,k → 0, whichequals the Mott formula. This also demonstrates that thefree Coulomb wave function assumed in eq. (1) is suitablefor describing the scattering of electrons by Ne-like argonas provided in the example. For larger angles, there arediscrepancies caused by the fact that the slow scatteringelectron probes the bound structure of the ion, which isnot taken into account by the Mott formula, as underlinedin introducing eq. (2). Nevertheless, such discrepanciescan be removed if Hartree-Fock calculations are used [45].

In fig. 2, we show the atomic FF of a Ne-like argontarget in the presence of a soft X-ray ER of wavelength≃ 6.2 nm, which is still higher than the first single-photonabsorption line. The atomic FF of Ne-like argon has beenanalytically calculated using the novel technique presentedin ref. [46]. As we see from fig. 2, the shift brought to theFF by the ER is of the order of few percent, and it de-pends on the two variables τ and s. We remark that,in order to have a probability ratio dσs =0/dσs=0 ∼ 1 or

τ τ π

τ τ π

τ π τ π/2

Fig. 2: (Color online) Atomic form factor of Ne-like argon in thepresence of ER with wavelength ≃ 6.2 nm. The parameter τrepresents the angle between the electron-target linear momen-tum transfer and the ER direction, i.e. cos τ = k ·Q/(|k||Q|).For the parameter s, see fig. 1.

Fig. 3: (Color online) Atomic form factor of a Xe atom in thepresence of ER with wavelength ≃ 6.2 nm. See figs. 2 and 1 forthe parameters τ and s.

greater with soft X-ray ER, radiation intensities as high asI 1018 W/cm2 must be employed, which might generatetwo-photon absorption peaks. Even so, since the first ion-ization threshold of Ne-like Ar is at ∼ 2.94 nm, two-photonabsorption may not cause ionization of the target. Suchintensities in the soft-X ray regime are nowadays achiev-able by using XFEL sources [26–28].

The investigation of the linear momentum shift in theFF of neutral samples can also be done, although samecare is needed. In order to obtain a considerable shift inneutral atoms, one needs high-intensity radiation whosephotons have energy higher than few hundreds electon-volts. Under these conditions neutral samples will un-dergo several ionizations, leading to numerous backgroundevents. However, control parameters can be used to mon-itor the sample damage, as done in crystallography for

13002-p4

Quantized form factor shift in the presence of free electron laser radiation

monitoring the crystal damage [47]. Alternatively, onemay use FEL sources and molecular structures. In fact, ithas been proved very recently that FEL prevents sampledamage in protein structures by simply outrunning it [48].In view of these considerations, we present in fig. 3 theshift brought to the FF of neutral Xe by soft X-ray ERof wavelength ≃ 6.2 nm (same as in fig. 2). The FF shiftis larger than in fig. 2, although still of the order of somepercent. The choice of the Xe atom is motivated by a re-cent LAES experiment carried out using fs pulses withwavelength of about 800 nm, intensity 1012 W/cm2 [21].

The problem of sample damage on a neutral samplemight be also circumvented by further increasing the ERintensity so as to reach a new stability regime (stabilitygiven by super-intense laser pulses) [49,50].

Remarks. – Linear momentum shifts in form factors ofclusters and mesoscopic objects can also be studied withthe very same formalism developed here [51–54]. A FFshift of the order ∼ eV/c is already significant in suchsystems, due to their larger size. This can be achieved byusing slow electrons and visible light.

As for the case of the KWF [55], the present formalismmay be extended to a relativistic framework by employingVolkov solutions and the full electron-target electromag-netic interaction [34]. Such an extension should permitto measure electric and magnetic FFs values at zero ar-gument from scattering events whose momentum transferis not zero. This would help solve the puzzle arisen fromrecent measurements on the proton structure, where it hasbeen showed that the ratio between electric and magneticFFs, at Q ≈ 0, is slightly less than what is to be ex-pected from QCD considerations [56,57]. The presentedformalism might be also employed to shed light onto re-cent disputed measurements of the proton radius [29,58].

Finally, form factor shifts can be also studied withboth radiation-free [59] and radiation-assisted [60] twistedelectrons. Such studies will be addressed in futurepublications.

∗ ∗ ∗

FF acknowledges support by Fundacao de Amparoa Pesquisa do estado de Minas Gerais (FAPEMIG)and Conselho Nacional de Desenvolvimento Cientıficoe Tecnologico (CNPq). LS, KJ and FF acknowledgesupport by the Research Council for Natural Sciencesand Engineering of the Academy of Finland. AGH ac-knowledges the support from the GSI Helmholtzzentrumand the University of Heidelberg. PA acknowledgessupport by the German Research Foundation (DFG)within the Emmy Noether program under Contract No.TA 740 1-1. JPS and PA acknowledge support by FCT—Fundacao para a Ciencia e a Tecnologia (Portugal),through the Projects No. PEstOE/FIS/UI0303/2011and PTDC/FIS/117606/2010, financed by the Eu-ropean Community Fund FEDER through theCOMPETE —Competitiveness Factors Operational

Programme. FF is thankful to Prof. Marcelo Franca

Santos, Prof. Sergio Scopetta, Dr. Wei Cao

and Dr. Abraham Kano for useful discussions andcomments.

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13002-p6

1

Supplementary material: Mathematical details

The amplitude for the scattering process in Eq. (2) of the text can be written in full glory as

A =(−i~)−1

V

∫ t

−t

dt′∫

d3r[e−

i~ t′(Ei−Ef )e

i~r(Pi−Pf )e−i(αi−αf )e−i(αi−αf ) cos[kr−ωt′]e−ikr(βi−βf )

× eiωt′(βi−βf )eiβi−βf

2 sin[2(kr−ωt′)] V (r)

].

(1)

On the equation above we shall use the Jacobi-Anger expansions. After having integrated over dt′, from Eq. (1) wefind

A =ei(αi−αf )

V

∑s,s1

Js(αi − αf )Js1

(βi − βf

2

)e−isπ

2

ϵ

(−2i sin

[tϵ

~

])∫d3r e

i~rχ V (r) , (2)

where ϵ = Ei −Ef + ~ω(βf − βi + s+ 2s1) and χ = Pi −Pf + ~k(βf − βi + s+ 2s1). We rewrite the integral aboveby making use of the equivalence∫

d3r ei~rχ V (r) =

∫d3r

(− ~2

χ2

) (∇2e

i~rχ

)V (r) = − ~2

χ2

∫d3r e

i~rχ

(∇2V (r)

)=

(−4π~3cαe

χ2

)∫d3re

i~rχ

(ϕ∗f (r)ϕi(r)

) (3)

where we used ∇2 1|r−r′| = 4πδ3(r − r′), and where αe is the electromagnetic coupling constant. In writing Eq. (3),

we have discarded the surface term coming from the partial integration, since the potential V (r) and its derivativeare vanishing for r → +∞. Joining Eq. (3) with Eq. (2), we readily get

A =(+8iπ~3cαe

) ei(αi−αf )

V

∑s,s1

Js(αi − αf )Js1

(βi − βf

2

)e−isπ

2

ϵχ2sin

[tϵ

~

]F (χ) , (4)

where F (χ) =

∫d3re

i~rχ

(ϕ∗f (r)ϕi(r)

). In order to obtain the differential probability for the process (dO), we take

the modulus squared of the amplitude and multiply it by the density of final states: dO ≡ |A|2 V d3Pf

(2π~)3 . The scattering

rate (dW ) is then obtained by taking the first derivative of dO with respect to the scattering time interval ∆t = 2t,and by sending ∆t → +∞, for obtaining energy conservation. Consequently, the oscillatory terms with differentfrequencies give zero contribution and after a few algebraic passages we are left with

dW =4~2c2α2

e

√2Efm

3/2

V

∑s,s1

(Js(αi − αf )Js1

(βi − βf

2

))2 |F (χ)|2

χ4δ(ϵ) dEfdΩPf (5)

where we used lim∆t→+∞sin[ ϵ∆t

~ ]ϵ/~ = πδ

(ϵ/~

)= π~δ

(ϵ)

and d3Pf = m3/2√2EfdEfdΩPf

. Here, dΩPfis the angle

differential of the vector Pf . In order to obtain the differential cross section (dσ), we must normalize the scatteringrate to the initial electron flux [1]. The initial electron flux can be written as J =< va > /V , where < va > is theaverage initial velocity of the electron beam along the scattering axis (i.e., the axis between the electron at initialtime and the target). The quantity < va > can be calculated from

< va >=1

m

∫d3rΨ∗

Pi,k(r)paΨPi,k(r) =

1

m(P a

i − ~βika) , (6)

where pa is the linear momentum operator along the scattering axis, and P ai , ka are the component along the

scattering axis of the electron initial linear momentum and of the radiation wave-vector, respectively. For moderatelaser intensities (or by assuming the radiation direction to be orthogonal to the scattering axis), we can approximateP ai − ~βik

a ≈ P ai = |Pi|, where the last step follows from the fact that the electron initial linear momentum is

normally along the scattering axis. With the help of Eq. (6) and by approximating βi − βf ≈ 0, we may finally write

dσ

dEfdΩPf

=1

J

dW

dEfdΩPf

= 4~2c2α2em

2 |Pf ||Pi|

+∞∑s=−∞

J2s (αi − αf )

|F (Q+ s~k)|2

(Q+ s~k)4δ(Ei − Ef + ~ωs

), (7)

2

where Q = Pi − Pf is the momentum transferred in the scattering process. By integrating over the energy of thescattered electron and by selecting the parameter s [2], we find Eq. (4) of the text.

[1] B. Povh, K. Rith, C. Scholz, and F. Zetsche, Particles and nuclei: an introduction to physical concepts (Springer, 2006),chapter 5.

[2] This can be done by monitoring the kinetic energy of the scattered electron.